3.576 \(\int \frac{c+d x^{n/2}+e x^n+f x^{3 n/2}}{\left (a+b x^n\right )^2} \, dx\)

Optimal. Leaf size=162 \[ \frac{x (a e-b c (1-n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 b n}-\frac{x^{\frac{n+2}{2}} (b d (2-n)-a f (n+2)) \, _2F_1\left (1,\frac{1}{2} \left (1+\frac{2}{n}\right );\frac{1}{2} \left (3+\frac{2}{n}\right );-\frac{b x^n}{a}\right )}{a^2 b n (n+2)}+\frac{x \left (x^{n/2} (b d-a f)-a e+b c\right )}{a b n \left (a+b x^n\right )} \]

[Out]

(x*(b*c - a*e + (b*d - a*f)*x^(n/2)))/(a*b*n*(a + b*x^n)) - ((b*d*(2 - n) - a*f*
(2 + n))*x^((2 + n)/2)*Hypergeometric2F1[1, (1 + 2/n)/2, (3 + 2/n)/2, -((b*x^n)/
a)])/(a^2*b*n*(2 + n)) + ((a*e - b*c*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 +
n^(-1), -((b*x^n)/a)])/(a^2*b*n)

_______________________________________________________________________________________

Rubi [A]  time = 0.264256, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114 \[ \frac{x (a e-b c (1-n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 b n}-\frac{x^{\frac{n+2}{2}} (b d (2-n)-a f (n+2)) \, _2F_1\left (1,\frac{1}{2} \left (1+\frac{2}{n}\right );\frac{1}{2} \left (3+\frac{2}{n}\right );-\frac{b x^n}{a}\right )}{a^2 b n (n+2)}+\frac{x \left (x^{n/2} (b d-a f)-a e+b c\right )}{a b n \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^(n/2) + e*x^n + f*x^((3*n)/2))/(a + b*x^n)^2,x]

[Out]

(x*(b*c - a*e + (b*d - a*f)*x^(n/2)))/(a*b*n*(a + b*x^n)) - ((b*d*(2 - n) - a*f*
(2 + n))*x^((2 + n)/2)*Hypergeometric2F1[1, (1 + 2/n)/2, (3 + 2/n)/2, -((b*x^n)/
a)])/(a^2*b*n*(2 + n)) + ((a*e - b*c*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 +
n^(-1), -((b*x^n)/a)])/(a^2*b*n)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 39.1909, size = 122, normalized size = 0.75 \[ - \frac{x \left (2 a e - 2 b c + 2 x^{\frac{n}{2}} \left (a f - b d\right )\right )}{2 a b n \left (a + b x^{n}\right )} + \frac{x \left (a e - b c \left (- n + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{2} b n} + \frac{x^{\frac{n}{2} + 1} \left (a f \left (n + 2\right ) - b d \left (- n + 2\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{n + 2}{2 n} \\ \frac{3}{2} + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{2} b n \left (n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c+d*x**(1/2*n)+e*x**n+f*x**(3/2*n))/(a+b*x**n)**2,x)

[Out]

-x*(2*a*e - 2*b*c + 2*x**(n/2)*(a*f - b*d))/(2*a*b*n*(a + b*x**n)) + x*(a*e - b*
c*(-n + 1))*hyper((1, 1/n), (1 + 1/n,), -b*x**n/a)/(a**2*b*n) + x**(n/2 + 1)*(a*
f*(n + 2) - b*d*(-n + 2))*hyper((1, (n + 2)/(2*n)), (3/2 + 1/n,), -b*x**n/a)/(a*
*2*b*n*(n + 2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.32216, size = 151, normalized size = 0.93 \[ \frac{x \left ((n+2) \left (a \left (b \left (c+d x^{n/2}\right )-a \left (e+f x^{n/2}\right )\right )+\left (a+b x^n\right ) (a e+b c (n-1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )\right )+x^{n/2} \left (a+b x^n\right ) (a f (n+2)+b d (n-2)) \, _2F_1\left (1,\frac{1}{2}+\frac{1}{n};\frac{3}{2}+\frac{1}{n};-\frac{b x^n}{a}\right )\right )}{a^2 b n (n+2) \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^(n/2) + e*x^n + f*x^((3*n)/2))/(a + b*x^n)^2,x]

[Out]

(x*((b*d*(-2 + n) + a*f*(2 + n))*x^(n/2)*(a + b*x^n)*Hypergeometric2F1[1, 1/2 +
n^(-1), 3/2 + n^(-1), -((b*x^n)/a)] + (2 + n)*(a*(b*(c + d*x^(n/2)) - a*(e + f*x
^(n/2))) + (a*e + b*c*(-1 + n))*(a + b*x^n)*Hypergeometric2F1[1, n^(-1), 1 + n^(
-1), -((b*x^n)/a)])))/(a^2*b*n*(2 + n)*(a + b*x^n))

_______________________________________________________________________________________

Maple [F]  time = 0.139, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( a+b{x}^{n} \right ) ^{2}} \left ( c+d{x}^{{\frac{n}{2}}}+e{x}^{n}+f{x}^{{\frac{3\,n}{2}}} \right ) }\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d*x^(1/2*n)+e*x^n+f*x^(3/2*n))/(a+b*x^n)^2,x)

[Out]

int((c+d*x^(1/2*n)+e*x^n+f*x^(3/2*n))/(a+b*x^n)^2,x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{{\left (b d - a f\right )} x x^{\frac{1}{2} \, n} +{\left (b c - a e\right )} x}{a b^{2} n x^{n} + a^{2} b n} + \int \frac{2 \, b c{\left (n - 1\right )} + 2 \, a e +{\left (a f{\left (n + 2\right )} + b d{\left (n - 2\right )}\right )} x^{\frac{1}{2} \, n}}{2 \,{\left (a b^{2} n x^{n} + a^{2} b n\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^(3/2*n) + d*x^(1/2*n) + e*x^n + c)/(b*x^n + a)^2,x, algorithm="maxima")

[Out]

((b*d - a*f)*x*x^(1/2*n) + (b*c - a*e)*x)/(a*b^2*n*x^n + a^2*b*n) + integrate(1/
2*(2*b*c*(n - 1) + 2*a*e + (a*f*(n + 2) + b*d*(n - 2))*x^(1/2*n))/(a*b^2*n*x^n +
a^2*b*n), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{f x^{\frac{3}{2} \, n} + d x^{\frac{1}{2} \, n} + e x^{n} + c}{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^(3/2*n) + d*x^(1/2*n) + e*x^n + c)/(b*x^n + a)^2,x, algorithm="fricas")

[Out]

integral((f*x^(3/2*n) + d*x^(1/2*n) + e*x^n + c)/(b^2*x^(2*n) + 2*a*b*x^n + a^2)
, x)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c+d*x**(1/2*n)+e*x**n+f*x**(3/2*n))/(a+b*x**n)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f x^{\frac{3}{2} \, n} + d x^{\frac{1}{2} \, n} + e x^{n} + c}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^(3/2*n) + d*x^(1/2*n) + e*x^n + c)/(b*x^n + a)^2,x, algorithm="giac")

[Out]

integrate((f*x^(3/2*n) + d*x^(1/2*n) + e*x^n + c)/(b*x^n + a)^2, x)